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Results tagged with ct.category-theory
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user 58211
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
2
votes
0
answers
128
views
Generalization of monoidal category with tensor products of $n$ objects
I'm looking for a generalization of monoidal categories, say $n$-monoidal categories, s.t. an ordinary monoidal category is the $n=2$ case. For general $n$, naively it should consist (among other data …
- 1,527
2
votes
1
answer
194
views
Appropriate morphisms and 2-morphisms in Ind(C)
As I was trying to understand the category $Ind(C)$ of diagrams of the form $I \to C$, where $I$ is a small filtered $(0,1)$-category, I wondered whether it is possible to define morphisms directly, w …
- 1,527
2
votes
0
answers
117
views
Grothendieck groupoid associated to bicategory
Given a finite abelian category $\mathcal{C}$, we can associate to $\mathcal{C}$ its Grothendieck group $\mathsf{Gr}(\mathcal{C})$, which is the free abelian group generated by isomorphism classes of …
- 1,527
3
votes
1
answer
218
views
Question on Eilenberg-Watts theorem
I'm not sure if this is a research level question, but:
Let $F:Rep_A \to Rep_B$ be an exact cocomplete functor between representation categories of finite dimensional $k$ algebras, where $k$ has char …
- 1,527
2
votes
0
answers
98
views
Simplicial basis in iterated bar construction
Let $G$ be an abelian group and set $A:=\mathbb{Z}G$. We can define a commutative dga Hopf algebra
$$B(A):=\bigoplus_{k \in \mathbb{Z}}\,B_k,$$
where $B_k:=A^{\otimes(k+1)}$. I like to think of the ba …
- 1,527
2
votes
0
answers
138
views
Coend of full subcategory
$\require{AMScd}$Let $F:\mathcal{C}^{op}\times \mathcal{C} \to \mathcal{D}$ be a functor and $\mathcal{C}' \subseteq \mathcal{C}$ a full subcategory. Assume that the coends $C$ over $F$ and $C'$ over …
- 1,527
5
votes
0
answers
238
views
Lie algebras in braided monoidal categories
Let $\mathcal{C}$ be a braided (not necessarily symmetric) monoidal category. Then we can define what monoids and commutative monoids in $\mathcal{C}$ are. What is the correct definition of a Lie alge …
- 1,527
4
votes
1
answer
358
views
Does the functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ have adjoints?
Let $\mathcal{C}$ be a braided monoidal category. We have a canonical functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ from $\mathcal{C}$ to the Drinfeld center $\mathcal{Z}(\mathcal{C})$ sending an …
- 1,527
8
votes
0
answers
207
views
Categorical interpretation of quantum double $D(A,B,\eta)$
It is known that the Drinfel'd double $D(A)$ of a Hopf algebra $A$ is characterized by the following two properties:
The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the ca …
- 1,527
4
votes
0
answers
145
views
Hopf monoid from comonoidal structures
Let $\mathcal{V}$ be a closed braided monoidal category and $\mathcal{V}-Cat$ the monoidal bicategory of small $\mathcal{V}$-enriched categories. Let $\mathcal{C}$ be a pseudo-comonoid in $\mathcal{V} …
- 1,527
4
votes
0
answers
90
views
Tensor algebras in the bicategory $\mathsf{2Vect}$
To my knowledge there are two main approaches to categorify the notion of a vector space. I will refer to them as BC-2-vector spaces (Baez, Crans) and KV-2-vector spaces (Kapranov, Voevodsky). Both de …
- 1,527
4
votes
0
answers
66
views
Categorical construction of comodule category of FRT algebra
Let $\mathcal{B}$ denote the braid groupoid, with objects being non-negative integers $n \in \mathbb{Z}_{\geq 0}$ and morphisms $\mathcal{B}(n,n)=B_{n}$ given by the braid group. Let $\mathcal{C}$ be …
- 1,527
12
votes
1
answer
527
views
Vertex algebras and factorization algebras
It is often said that vertex algebras are a special case of factorization algebras. In particular, in their book "FAs in QFT" Costello/Gwilliam construct a functor from a certain class of 2d "holomorp …
- 1,527
1
vote
0
answers
82
views
Braided category inside braided 2-category
Let $\mathcal{C}$ be a semistrict braided monoidal $2$-category in the sense of [BN] (so in particular a strict $2$-category). Let $\mathcal{C}_1$ be the category of $1$-morphisms (objects) and $2$-mo …
- 1,527